Misconceptions arising from the Infinite Solenoid Magnetic Field Formula

Abstract

Many high school and first-year university courses include discussion of the magnetic effect of currents. Frequently discussed textbook examples include long, straight wires, circular current loops, and solenoids, partly because these examples are tractable mathematically. The solenoid naturally leads to discussion on magnetic materials since it is readily demonstrated that a paramagnetic core significantly boosts the strength of an electromagnet. However, magnetic effects of solid and even liquid materials are subtle and confusing¹ and the mathematics is not straightforward. This leads to confusion amongst students (and their teachers), which, when taken to more advanced study, leads to significant misconceptions about the nature of magnetic properties and fields. These misconceptions can become problematic when practical (rather than stereotyped) magnetic design and analysis is required such as for transformers,² magnetic recording materials, geomagnetic sensors,³ or biological stimulators4 to name a few. In this article, I highlight examples of this confusion, in particular the failure in realistic situations of the well-quoted formula for an infinite solenoid with a paramagnetic core, and the physical interpretation of the relative permeability of a material, µr.